There are a few points here. (1) Why use a reciprocal lattice? (2) What does it represent? (3) What information can you get directly from reciprocal space without having to transform back to direct ("real") space.

(1) The defining quality of crystals is their periodicity. The natural way to study periodic things is to Fourier transform them.

This probably does not sound very convincing to students unless you can find a few examples they are familiar with. Outside of physics, frequency is almost always associated with time rather than space (audio, radio, etc), and a 3D FT will scare beginners.

(2) The FT of the crystal (duh). It is worth recalling that small Q in reciprocal space represent large distances in direct space and vice versa.

Roughly, what is inside the BZ represents how one unit cell relates to the next.
On the other hand, in crystallography one measures the amplitude of many reciprocal lattice points to reconstruct what's inside the RS unit cell

(3) Loads of things. Band structure of course. Crystallography, Phonons, the BCS theory of superconductivity.

Like M Quack said, reciprocal lattice is a mathematical structure which allows the application of analytic geometry of linear forms to coordinate systems with arbitrary bases, including the non-orthonormal bases of lattices with low symmetry. Makes many calculations possible! You can also show your students diffraction experiments and explain the Bragg condition for constructive interference on a lattice, as it's done in Kittel's Introduction to solid state physics, afair.

Suz85, G's are simply basis vectors of the reciprocal lattice, while k is any (continuous) vector in the reciprocal space.